## Does catastrophe theory represent a major development in mathematics

MATHEMATICS AND COMPUTER SCIENCE

# Does catastrophe theory represent a major development in mathematics?

**Viewpoint:** Yes, catastrophe theory is a major development in mathematics.

**Viewpoint:** No, catastrophe theory does not represent a major development in mathematics: not only is it a close cousin of chaos theory, but both are products of a twentieth-century intellectual environment that introduced quantum mechanics and other challenges to a traditional "common sense" worldview.

The twentieth century saw the introduction of complexities in ways no previous era could have imagined: complexities in political systems, in art, in lifestyles and social currents, and most of all, in the sciences and mathematics. Leading the movement that increasingly unfolded the complexities at the heart of the universe were quantum mechanics in physics and chemistry, and relativity theory in physics. As for the mathematical expressions of complexity, these were encompassed in a variety of concepts known collectively (and fittingly enough) as *complexity theory*. Among the various mathematical approaches to complexity are chaos theory and the closely related mathematical theory of catastrophe, originated by French mathematician René Thom (1923-).

There are several varieties of misunderstanding surrounding the ideas of complexity, chaos, and catastrophe. First of all, the umbrella term *complexity* seems to suggest complexity for complexity's sake, or an exclusive focus on things that are normally understood as complex, and nothing could be further from the truth. In fact, complexity theory—particularly in the form known as *catastrophe theory*—is concerned with events as apparently simple as the path a leaf makes in falling to the ground. As complexity theory shows, however, there is nothing simple at all about such events; rather, they involve endless variables, which collectively ensure that no two leaves will fall in exactly the same way.

There is nothing random in the way that the leaf falls; indeed, quite the contrary is the case, and this points up another set up misconceptions that arise from the names for various types of complexity theory. Chaos theory is not about chaos in the sense that people normally understand that term; in fact, it is about exactly the opposite, constituting a search for order within what appears at first to be disorder. In attempting to find an underlying order in the seemingly random shapes of clouds, for instance, chaos theory is naturally concerned with complexities of a magnitude seldom comprehended by ordinary mathematics.

Likewise, catastrophe theory does not necessarily involve "catastrophes" in the sense that most people understand that word, though many of the real-world applications of the idea are catastrophic in nature. In the essays that follow, catastrophe theory is defined in several different ways, and represented by numerous and diverse examples. At heart, all of these come back to a basic idea of sudden, discontinuous change that comes on the heels of regular, continuous processes. An example—and one of the many "catastrophic" situations to which catastrophe theory is applied—would be the sudden capsizing of a ship that has continued steadily to list (tilt) in a certain direction.

At issue in the essays that follow is the question of whether catastrophe theory represents a major development in mathematics. To an extent, this is a controversy that, like many in the realm of mathematics or science, takes place "above the heads" of ordinary people. Most of us are in the position of an unschooled English yeoman of the eighteenth century, poking his head through a window of Parliament to hear members of the House of Lords conducting a debate concerning affairs in India—a country the farmer has never seen and scarcely can imagine.

Yet just as England's position vis-á-vis India *did* ultimately affect the farmer, so our concern with catastrophe mathematics represents more than an academic interest. If catastrophe theory does represent a major new movement, then it opens up all sorts of new venues that will have practical applications for transportation, civil engineering, and other areas of endeavor that significantly affect the life of an ordinary person. On the other hand, if catastrophe theory is not a significant new development, then this is another blow against complexity theory and in favor of the relatively straightforward ideas that dominated mathematical thought prior to the twentieth century.

Catastrophe theory has a long and distinguished history, with roots in the sixteenth-century investigations of Leonardo da Vinci (1452-1519) into the nature of geometric patterns created by reflected and refracted light. Serious foundational work can be found in the bifurcation and dynamical systems theory of French mathematician Jules Henri Poincaré (1854-1912). Yet the heyday of catastrophe as a mathematical theory was confined to a period of just a few years in the third quarter of the twentieth century: thus, by the time its originator had reached the mere age of 55, even supporters of his theory were already proclaiming catastrophe a dead issue. Thom himself, in a famous statement quoted by both sides in the controversy at hand, stated that "catastrophe theory is dead."

If both sides concede that catastrophe theory is dead as an exciting mathematical discipline, then one might wonder what question remains to be decided. But it is precisely the meaning of this "death" that is open to question. Seeds die, after all, when they are planted, and the position of catastrophe supporters in this argument is that catastrophe is dead only in a formal sense; far from being truly dead, it flourishes under other names. Detractors, on the other hand, deride catastrophe theory as a mere gimmick, a sophisticated parlor trick that has been misapplied in sometimes dangerous ways to analyses of social science.

Both sides, in fact, agree that the use of catastrophe theory for analyzing social problems is questionable both in principle and in application. Adherents of catastrophe theory as a bold new mathematical idea are particularly strong on this point, taking great pains to differentiate themselves from those who at various times have sought to use catastrophe as a sort of skeleton key to unlocking underlying secrets of mass human behavior.

Again, however, there is a great deal of misunderstanding among the general public with regard to the ways mathematical theories are used in analyzing social problems. The mere use of mathematical models in such situations may strike the uninitiated as a bit intrusive, but in fact governments regularly depend on less controversial mathematical disciplines, such as statistics, in polling the populace and planning for the future.

A final area of agreement between both sides in the catastrophe dispute is a shared admiration for the genius of Thom and for the work of other mathematicians—most notably Dutch-born British mathematician Christopher Zeeman (1925-)—who helped develop the theory as a coherent system. When it comes to catastrophe theory, evidence does not support the assertion that "to know it is to love it." However, even those who would dismiss catastrophe as mere mathematical trickery still admit that it is an extraordinarily clever form of trickery.

—JUDSON KNIGHT

## Viewpoint: Yes, catastrophe theory is a major development in mathematics.

Not only is catastrophe theory a major development in mathematics, it also has many uses in other fields as diverse as economics and quantum physics. Catastrophe theory is itself a development of topology, and builds on the work of earlier mathematical concepts. The only controversy surrounding the status of catastrophe theory is not a mathematical one, but rather over some of the non-mathematical uses to which it has been applied.

Catastrophe theory attempts to describe situations where sudden discontinuous results follow from smooth, continuous processes. This puts it outside the framework of traditional calculus, which studies only smooth results. The real world is full of discontinuous jumps, such as when water boils, ice melts, the swarming of locusts, or a stock market collapse. Catastrophe theory is an attempt to offer a geometric model for such behaviors.

The theory can be described in a number of ways: as the study of the loss of stability in a dynamic system, the analysis of discontinuous change, a special subset of topological forms, or even a controversial way of thinking about transformations. However, the most accurate and unambiguous way of describing catastrophe theory is by using mathematical terminology. Unfortunately, even the most basic mathematical understanding requires knowledge of calculus in several variables and some linear algebra. Yet, by virtue of its initial popularity, many books have been written on catastrophe theory for the non-mathematician. While a good number of these do a wonderful job of describing the nature of catastrophe theory, giving easy-to-understand examples and infusing the reader with enthusiasm for the topic, they also, by their very nature, simplify the subject. They tend to stress the non-mathematical applications of the theory, and that is when the controversy begins.

### Catastrophe Theory and the Media

The media does not usually take notice of the release of a mathematics text. Yet the English translation of René Thom's work *Structural Stability and Morphogenesis: An Essay on the General Theory of Models*, published originally in French in 1972, was widely reviewed. It not only spawned dozens of articles in the scientific press, but also in such mainstream publications as *Newsweek*. The theory's name was partly to blame. The word "catastrophe" evoked images of nuclear war and collapsing bridges, and the media suggested the theory could help explain, predict, and avoid such disasters.

Yet the aim of the theory was altogether different, if almost as ambitious. Thom had a long track record of pure mathematical work in topology, including characteristic classes, cobordism theory, and the Thom transversality theorem (for which he was awarded a Fields medal in 1958). However, it was while thinking about the biological phenomena of morphogenesis that he began to see some topological curiosities in a broader light. Biological morphogenesis is the astonishing way in which the small collection of embryonic cells suddenly specialize to form arms, legs, organs, and all the other parts of the body. Such sudden changes fascinated Thom, and he began to see such transformations in form everywhere, from the freezing of water into ice, to the breaking of waves. He began formulating a mathematical language for these processes, and with the help of other mathematicians, such as Bernard Malgrange, he reached a remarkable conclusion. In 1965 Thom discovered seven elementary topological forms, which he labeled "catastrophes" for the manner in which small changes in the parameters could produce sudden large results. These elementary catastrophes, Thom argued, were like the regular solids and polygons in Greek geometry. Despite the infinite number of sides a polygon can have, only three regular polygons—the triangle, square, and hexagon—can be placed in edges to fill a surface such as a tiled floor. Also, there are only five regular solids that can be made by using regular polygons as the sides. These forms recur again and again in nature, from honeycombs to crystals, precisely because they are the only ways to fill a plane surface, or form a regular shape.

Thom suggested that the seven elementary catastrophes also recur in nature in the same way. This offers a new way of thinking about change, whether it be a course of events or a dynamic system's behavior, from the downfall of an empire to the dancing patterns of sunlight on the bottom of a swimming pool.

Thom left much of the mathematical "dirty work" to others, and so it was John Mather who showed that the seven abstract forms Thom called elementary catastrophes existed, were unique, and were structurally stable. Indeed, Thom's methods were one of the reasons catastrophe theory was initially criticized. Thom's mathematical work has been liken to that of a trailblazer, hacking a rough path through a mathematical jungle, as opposed to other, more rigorous mathematicians, who are slowly constructing a six-lane highway through the same jungle. Even one of Thom's most ardent followers noted that "the meaning of Thom's words becomes clear only after inserting 99 lines of your own between every two of Thom's."

### Wider Applications

However obscurely phrased, Thom's ideas quickly found an audience. In a few years catastrophe theory had been used for a wide variety of applications, from chemical reactions to psychological crises. It was the work of E. Christopher Zeeman, in particular, that was most provocatively reported. Zeeman had attempted to apply the theory in its broadest sense to many fields outside of mathematics. Zeeman modeled all kinds of real-world actions he argued had the form of catastrophes, from the stability of ships to the aggressive behavior of dogs. Perhaps his most controversial model was that of a prison riot at Gartree prison in England. The study was challenged on the grounds that it was an attempt to control people in the manner of Big Brother, and some media reports incorrectly stated that hidden cameras were being used to invade the privacy of the test subjects. In reality the modeling used data obtained before the riot by conventional means that was available in the public domain.

While new supporters of the theory quickly applied it to an ever-growing list of disciplines, from physics to psychology, others argued that catastrophe theory offered nothing of value, used *ad hoc* assumptions, and was at best a curiosity, rather than a universal theory of forms as Thom had suggested. Indeed, some saw it not just as insignificant, but as a danger. Hector Sussmann and Raphael Zalher criticized it as "one of the many attempts that have been made to deduce the world by thought alone," and Marc Kac went so far as to charge that catastrophe theory was "the height of scientific irresponsibility."

Some of the applications were easy targets for the critics. For example, political revolutions were modeled by Alexander Woodcock using the two-variable cusp elementary catastrophe. The variables chosen were political control and popular involvement. The "results" of mapping flow lines onto the topological surface of the cusp catastrophe suggested some controversial conclusions. High popular involvement in politics would always seem to bring a dictatorship to a catastrophe point, yet many historians would argue that sometimes high popular involvement helps stabilize a dictatorship, such as the early years of the Nazi regime. Some critics suggested that the chosen variables were not as important as other factors, for example, economic performance, or media perception.

### A New Tool, Not a Replacement

Yet the criticisms, while valid, miss the point of such models. They are not to be seen as complete explanations of real-world events, but merely attempts to offer new insights, however incomplete they might be. Competing models, using other variables, can be compared, and more sophisticated, higher-dimensional catastrophes can be used, although such four, five, and higher dimensional models lose their visual representation, being impossible to draw. Catastrophe theory is not a replacement for old methods, but rather a new supplement to test and compare other methods against. Even if the models have no validity at all in the long term, they still have the power to challenge conventional wisdom and offer new and exciting avenues of inquiry.

Part of the problem with the media perception of catastrophe theory is the difficulty of relating the mathematical details of the theory to a non-mathematical audience. Mathematically catastrophe theory is a special branch of dynamical systems theory, using differential topology. It attempts to describe situations in which gradually changing forces lead to abrupt changes. This is impossible using differential calculus, but that does not mean that catastrophe is a replacement for calculus, rather it is a new tool to use in situations previously unexplored by traditional mathematics. For example, calculus cannot predict the path of a leaf, or where it will land, as there are too many variables. However, there are patterns that can be deduced. No two falling leaves follow the same quantitative path, but they have a shared qualitative behavior that can be studied.

### Based on a Solid Mathematical Foundation

Catastrophe theory was not a mathematical revolution, but a development from the work of previous mathematicians. It builds on the theory of caustics, which are the bright geometric patterns created by reflected and refracted light, such as a rainbow or the bright cusp of light in a cup of coffee. Early work on caustics was done by Leonardo da Vinci (1452-1519), and their name was given by Ehrenfried Tschirnhaus (1651-1708). It also uses bifurcation theory, which was developed by Henri Poincaré (1854-1912) and others, and examines the forking or splitting of values. It also draws heavily on Hassler Whitney's (1907-1989) work on singularities. Indeed, without Whitney's developments the theory could not have been conceived. Thom's own earlier work on transversality was also important. The calculus of variations, which has a long and distinguished mathematical history, is also important to the theory. Catastrophe theory's development from such diverse fields shows that it is not something out of left field, but a novel way of combining many previous mathematical ideas and some new work on topological surfaces. In this way catastrophe theory is a major development in mathematics, even without considering any wider applications.

Catastrophe theory's basis in topology means that in general it will be concerned with qualities, not quantities. The elementary catastrophes are like maps with a scale. Models using them can only be approximate, and stress properties, not specific results. This means that such models cannot be used to predict events. Thom himself has noted that the scientific applications of his theory are secondary to its formal beauty and power, and its scientific status rests on its "internal, mathematical consistency," not the questionable models it has been used for.

### The Death of Catastrophe Theory?

Another criticism that has been leveled against catastrophe theory is that it quickly vanished. In just a few years, after the initial flurry of papers and attempted real-world applications the theory seemed to disappear from sight. Indeed, as early as 1978 supporters of catastrophe theory were already noting its demise. Tim Poston and Ian Stewart, in their book *Catastrophe Theory and its Applications*, remarked that "Catastrophe theory is already beginning to disappear. That is, 'catastrophe theory' as a cohesive body of knowledge with a mutually acquainted group of experts working on its problems, is slipping into the past, as its techniques become more firmly embedded in the consciousness of the scientific community." Thom himself stated, "It is a fact that catastrophe theory is dead. But one could say that it died of its own success." Rather than disappearing, catastrophe theory was absorbed into other disciplines and called by other, less controversial, names such as bifurcations and singularities. Catastrophe theory and its descendants have been particularly successful in fields already using complex mathematical analysis. From quantum physics to economics, catastrophe theory continues to be used in more and more innovative ways.

While catastrophe theory is an example of the normal development of mathematics, its importance should not be downplayed. Even the critics of catastrophe theory admit that the mathematics behind catastrophe theory is elegant and exciting. Their opposition to the theory is not a mathematical one, but rather with the non-mathematical uses the theory has been applied to. Like fractals and chaos theory, the ideas of catastrophe theory briefly captured popular attention. However, while the media may have suggested that catastrophe theory had the power to predict everything from stock market crashes to earthquakes, the reality of the theory is such that casual uses offer nothing more than interesting oddities. Even the most rigorous mathematical uses of catastrophe theory may return very little in terms of modeling the real world. However, the disappointments that the theory could not live up to the initial hype and exaggerations in no way diminishes the mathematical development that the theory represents.

—DAVID TULLOCH

## Viewpoint: No, catastrophe theory does not represent a major development in mathematics: not only is it a close cousin of chaos theory, but both are products of a twentieth-century intellectual environment that introduced quantum mechanics and other challenges to a traditional "common sense" worldview.

The mathematical principles known under the intriguing, though somewhat misleading, title of "catastrophe theory" are certainly not lacking in what an advertiser might call "sex appeal." Like the more well-known ideas of relativity theory or quantum mechanics, or even chaos theory (to which catastrophe theory is closely related), the precepts of catastrophe mathematics are at once intellectually forbidding and deceptively inviting to the uninitiated.

It is easy to succumb to the intoxicating belief that catastrophe theory is a major development in mathematics—that it not only signals a significant shift in the direction of mathematical study, but that it provides a new key to understanding the underlying structure of reality. This, however, is not the case. Catastrophe theory is just one of many intriguing ideas that emerged from the twentieth-century intellectual environment that introduced quantum mechanics and relativity, two theories that truly changed the way we view the world.

Relativity and quantum mechanics, of course, were and continue to be major developments in modern thought: almost from the time of their introduction, it was clear that both (and particularly the second of these) presented a compelling new theory of the universe that overturned the old Newtonian model. These ideas—and again, particularly quantum mechanics—served to call into question all the old precepts, and for the first time scientists and mathematicians had to confront the possibility that some of their most basic ideas were subject to challenge. The resulting paradigm shift, even as it signaled a turn away from the old verities of Newtonian mechanics, also heralded the beginnings of an intellectual revolution.

Chaos and its close relative, catastrophe theory, are the result of extraordinary insights on the part of a few creative geniuses working at the edges of mathematical discovery. But does either area represent a major development in mathematics, the opening of a new frontier? If either of the two were likely to hold that status, it would be the more broad-reaching idea of chaos theory, which Robert Pool analyzed in *Science* in 1989 with an article entitled "Chaos Theory: How Big An Advance?" The dubious tone of the title suggests his conclusion: paraphrasing Steven Toulmin, a philosopher of science at Northwestern University, Pool wrote, "Chaos theory gives us extra intellectual weapons, but not an entirely new worldview."

As for catastrophe theory, there is this pronouncement: "It is a fact that catastrophe theory is dead. But one could say that it died of its own success…. For as soon as it became clear that the theory did not permit quantitative prediction, all good minds … decided it was of no value." These words belong to French mathematician René Thom (1923-), originator of catastrophe theory, quoted by the theory's chief popularizer, Dutch-born British mathematician Christopher Zeeman (1925-).

### Complexity Theory

Though they arose independently, chaos and catastrophe theories can ultimately be traced back to the theory of dynamical systems, originated by French mathematician Jules Henri Poincaré (1854-1912). Whereas ordinary geometry and science are concerned with the variables that differentiate a particular entity, dynamical systems is devoted purely to addressing entities as systems moving through a particular state. Thus if the "clouds" of cream in a cup of coffee exhibit behavior similar to that of a hurricane, then this relationship is of interest in dynamical systems, even though a vast array of physical, chemical, and mathematical variables differentiate the two entities.

These three—catastrophe, chaos, and dynamical systems—are part of a larger phenomenon, which came into its own during the twentieth century, known as complexity theory. Also concerned with systems, complexity is based on the idea that a system may exhibit phenomena that cannot be explained by reference to any of the parts that make up that system. Complexity theory in general, and chaos theory in particular, involve extremely intricate systems in which a small change may yield a large change later. Weather patterns are an example of such a system, and in fact one of the important figures in the development of chaos theory was a mete-orologist, Edward Lorenz (1917-).

Specifically, chaos theory can be defined as an area of mathematical study concerned with non-linear dynamic systems, or forms of activity that cannot easily be represented with graphs or other ordinary mathematical operations. The name "chaos," which may have originated with University of Maryland mathematician James Yorke (1941-)—a leading proponent of the theory—is something of a misnomer. The systems addressed by chaos theory are the opposite of chaotic; rather, chaos theory attempts to find the underlying order in apparently random behavior.

### How Far Should We Take Catastrophe Theory?

Similarly, catastrophe theory is not really about catastrophes at all; rather, it is an area of mathematical study devoted to analyzing the ways that a system undergoes dramatic changes in behavior following continuous changes on the part of one or more variables. It is applied, for instance, in studying the way that a bridge or other support deforms as the load on it increases, until the point at which it collapses. Such a collapse is, in ordinary terms, a "catastrophe," but here the term refers to the breakdown of apparently ordered behavior into apparently random behavior. Again, however, there is the underlying principle that even when a system appears to be behaving erratically, it is actually following an order that can be discerned only by recourse to exceptionally elegant mathematical models, such as the catastrophe theory.

Catastrophe theory has proven useful in discussing such particulars as the reflection or refraction of light in moving water, or in studying the response of ships at sea to capsizing waves; however, its results are more dubious when applied to the social sciences. In the latter vein, it has been utilized in attempts to explain animals' fight-or-flight responses, or to analyze prison riots. One mathematician (though he will be remembered in quite different terms) even saw in catastrophe theory a model for the breakdown of Western society. His name: Theodore Kaczynski, a.k.a. the Unabomber, who in his infamous 1995 "manifesto" referred to the writings of Zeeman.

Catastrophe theory may be useful for analyzing aspects of the universe, but it is not a key
to the *entire* universe. Even relativity and quantum mechanics, much more far-reaching theories, only apply in circumstances that no one has directly experienced—i.e., traveling at the speed of light, or operating at subatomic level. In the everyday world of relatively big objects in which humans operate, old-fashioned Newtonian mechanics, and the mathematics necessary to describe it, still remain supreme. This suggests that neither catastrophe theory nor any other esoteric system of mathematical complexity is truly a major development in our understanding of the world.

—JUDSON KNIGHT

## Further Reading

Arnol'd, V. I. *Catastrophe Theory*. Berlin:Springer-Verlag, 1992.

Ekeland, Ivar. *Mathematics and the Unexpected*. Chicago: University of Chicago Press, 1988.

Hendrick, Bill. "Clues in Academia: Manifesto Steered Probe to 'Catastrophe Theory.'" *Journal-Constitution* (Atlanta) (April 6, 1996).

"Indexes of Biographies." School of Mathematics and Statistics, University of St. Andrews, Scotland. May 31, 2002 [cited July 16, 2002]. <http://www-groups.dcs.st-and.ac.uk/~history/BiogIndex.html>.

Pool, Robert. "Chaos Theory: How Big an Advance?" *Science* 245, no. 4913 (July 7, 1989): 26-28.

Poston, Tim, and Ian Stewart. *Catastrophe Theory and Its Applications*. London: Pitman, 1978.

Saunders, Peter Timothy. *An Introduction to Catastrophe Theory*. Cambridge, England: Cambridge University Press, 1980.

"Turbulent Landscapes: The Natural Forces That Shape Our World." *Exploratorium*. May 31, 2002 [cited July 16, 2002]. <http://www.exploratorium.edu/complexity/>.

Zeeman, Eric Christopher. *Catastrophe Theory—Selected Papers 1972-1977*. Reading, MA: Addison-Wesley, 1977.

## KEY TERMS

### CATASTROPHE THEORY:

An area of mathematical study devoted to analyzing the ways that a system undergoes dramatic changes in behavior following continuous changes on the part of one or more variables.

### CAUSTICS:

The word means "burning" and usually refers to light rays, and the intense, sharp, bright curves that form on certain surfaces; for example, the patterns of light in a cup of coffee, an empty saucepan, or a rainbow. They can be thought of as maximum points of intensity of the light.

### CHAOS THEORY:

An area of mathematical study concerned with non-linear dynamic systems, or forms of activity that cannot easily be represented with graphs or other ordinary mathematical operations.

### COMPLEXITY:

A mathematical theory based on the idea that a system may exhibit phenomena that cannot be explained by reference to any of the parts that make up that system.

### DYNAMICAL SYSTEMS:

A theory of mathematics concerned with addressing entities as systems moving through a particular state.

### MORPHOGENESIS:

The shaping of an organism by embryological processes of differentiation of cells, tissues, and organs.

Also used in a broader sense as any complex system-environment interaction that alters a system's form and structure. The most common example is the growth of an animal from a fertilized egg, where seemingly identical embryonic cells differentiate into the specialized parts of the body.

### SINGULARITY THEORY:

Generalization of the study of mathematical functions at the maximum and minimum points. Hassler Whitney introduced the topological notion of mappings, and concluded that only two kinds of singularities are stable, folds and cusps.

### SYSTEM:

In mathematics and the sciences, a system is any set of interactions that can be separated from the rest of the universe.

### TOPOLOGY:

The mathematics of geometrical structures. It is concerned with the qualities, not the specific quantities, of such structures, such as how many holes an object has. For example, topologically speaking a donut and a coffee mug are the same, as they have one hole. Another example is the London Underground Map, which does not have accurate directions and distances, but does show how all the lines and stations connect. Topology often deals with structures in multiple dimensions.

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